Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383548 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14361 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12)(4,17)(5,7,20,10)(8,16)(9,24)(13,21)(14,29,31,32)(15,23)(18,26)(19,25)(22,28,27,30);(2,7)(3,13)(4,18)(5,20)(10,11)(15,16)(17,25)(19,24)(21,23)(22,29)(27,32)(28,30);(1,3)(2,8)(4,14)(5,15)(6,16)(7,21)(9,22)(10,23)(11,12)(13,20)(17,32)(18,30)(19,31)(24,28)(25,29)(26,27);(1,4,5,18,6,19,20,24)(2,9,10,25,11,26,7,17)(3,14,15,30,16,31,13,28)(8,22,23,29,12,27,21,32)Polycyclic presentation:
2^9|a^2=h,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi,h^c=hi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251784, #72251792, #72251798, … |
| maximal normal subgroup of | #72251784, #72251792, #72251798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290744);